Find the new positions of the given triangle’s vertices after rotating it 180 degrees counterclockwise about 𝐿.
You can imagine that we’re going to put our finger on point 𝐿 and spin the lines 180 degrees counterclockwise. Before we go any further, let’s clarify what it means to go 180 degrees counterclockwise about a point, by looking at this point in yellow. If I want to take this point and turn this line 180 degrees counterclockwise, our yellow point would not change, but the line would go in the opposite direction.
Here’s what we should notice about our points. Point 𝐿 will not change. The 𝐿 prime or the new position of 𝐿 will be the same as its old position, five, four. Now, let’s look at 𝑀 compared to 𝐿. To get to 𝑀 from 𝐿, you go to the right three and up one. If we want a 180-degree counterclockwise position of 𝑀, we need to do the opposite of this. We should go three places left and one place down. This will be our new 𝑀, 𝑀 prime.
We have to do the same process with our 𝑁. 𝑁 is located three units up and one unit to the right of 𝐿. To go 180 degrees counterclockwise about 𝐿, we’ll need to do the opposite of three up and one to the right. We’ll go three units down and one unit left. This will give us the location of our new 𝑁, our 𝑁 prime.
Now, I can connect all of these points to see our new triangle. Here’s our new triangle. Again, you can imagine if you put your finger on the point at 𝐿, on that pink dot, you can see that this represents a 180-degree turn counterclockwise. We can’t forget that this question wants to know the vertices of all three points. So we still need to identify where 𝑀 prime and 𝑁 prime are on the graph.
What position is 𝑀 prime in? Two units to the right and three units up. 𝑀 prime is point two, three. 𝑁 prime is located at four units on the 𝑥-axis, four units to the right and one unit up. 𝑁 prime is found at point four, one. After we rotate our triangle 180 degrees counterclockwise about 𝐿, we have point 𝐿 prime at five, four, 𝑀 prime at two, three, and 𝑁 prime at four, one.